Question: Let $x$ be a real number, $x > 1.$  Compute
\[\sum_{n = 0}^\infty \frac{1}{x^{2^n} - x^{-2^n}}.\]
Answer: We can write
\[\frac{1}{x^{2^n} - x^{-2^n}} = \frac{x^{2^n}}{x^{2^{n + 1}} - 1}.\]Let $y = x^{2^n}.$  Then
\begin{align*}
\frac{x^{2^n}}{x^{2^{n + 1}} - 1} &= \frac{y}{y^2 - 1} \\
&= \frac{(y + 1) - 1}{y^2 - 1} \\
&= \frac{y + 1}{y^2 - 1} - \frac{1}{y^2 - 1} \\
&= \frac{1}{y - 1} - \frac{1}{y^2 - 1} \\
&= \frac{1}{x^{2^n} - 1} - \frac{1}{x^{2^{n + 1}} - 1}.
\end{align*}Thus, the sum telescopes:
\[\sum_{n = 0}^\infty \frac{1}{x^{2^n} - x^{-2^n}} = \left( \frac{1}{x - 1} - \frac{1}{x^2 - 1} \right) + \left( \frac{1}{x^2 - 1} - \frac{1}{x^4 - 1} \right) + \left( \frac{1}{x^4 - 1} - \frac{1}{x^8 - 1} \right) + \dotsb = \boxed{\frac{1}{x - 1}}.\]